Paul LarsonChoosing Ideals
We show that, in certain inner models of determinacy, there
is a definable procedure which, given a tall ideal $I$ on $\omega$
containing all finite sets, and a function from $I \setminus
\mathrm{Fin}$ to a countable set, chooses a finite subset of
the range of this function. In the case we are most interested
in, $I$ is generated by a countable collection of pairwise orthogonal
ideals. In this context, $I$ represents a $\mathcal{P}(\omega)/\mathrm{Fin}$-name
for an element of a countable set. Our result then says that,
whenever $M[U]$ is a $\mathcal{P}(\omega)/\mathrm{Fin}$-extension
of a model $M$ of the type we consider here, if $F$ is a function
in $M[U]$ with domain $X \in M$, and $G$ is a function in $M$
with domain $X$ such that, for all $x \in X$, $F(x) \in G(x)$
and $G(x)$ is countable, then there exists a function $G'$ in
in $M$ with domain $X$ such that, for all $x \in X$, $G'(x)$
is a finite subset of $G(x)$. This gives another proof of a
theorem of Di Prisco-Todorcevic that in $M[U]$ there is no function
which selects a single member from each $E_{0}$-equivalence
class, where $E_{0}$ is the relation of mod-finite agreement
on the Baire space.

June 3
Room 210

No Seminar

May 27 --1:30 to 3pm
Room 210

Jan van Mill (VU Amsterdam)Homeomorphism groups of homogeneous compacta need not
be minimalWe present an example of a homogeneous compact space,
the homeomorphism group of which is not minimal. This answers
a question of Stoyanov from about 1984. If time permits, we
will also talk about unique homogeneity and raise some open
problems.

Friday, May 20

No seminar

Friday, May 13 from 1:30 to
3pm
Fields Institute, Room 210

Peng Yinhe (Toronto
and Singapore)An L space with non-Lindelof square
There is an L space with non-Lindelof square. Moreover, there
is an L space with some stronger property that its square is
the closure of a countable union of closed discrete subsets.

We will discuss a few instances where the Vapnik-Chervonenkis
theory of statistical learning has potentially interesting
overlaps with set theory / model theory / extremal combinatorics.
This will be illustrated on speaker's recent results about
PAC learnability over non-atomic measures (a solution of a
problem by Vidyasagar with the help of Martin's Axiom), as
well as learnability over exchangeable data inputs, and a
reputed 25-year old open problem concerning the so-called
sample compression schemes.

An unprovable theorem is a theorem about basic mathematical
objects that can only be proved using more than the usual
axioms for mathematics (ZFC = Zermelo Frankel set theory with
the Axiom of Choice) - and that has been proved using standard
extensions of ZFC generally adopted by the mathematical logic
community.

The highlight of the talk is the presentation of unprovable
theorems stated in terms of self embeddings of maximal cliques
in graphs.

We first review some previous examples of unprovable theorems.
1-5 are unprovable in the weaker sense that any proof demonstrably
requires some use of logical principles transcendental to
the problem statement. 6 is BRT (Boolean Relation Theory).

1. Patterns in finite sequences from a finite alphabet.<br>
2. Pointwise continuous embeddings between countable sets
of reals (or more concretely, rationals).<br> 3. Relations
between f(n_1,...,n_k) and f(n_2,...,n_k+1).<br> 4.
Homeomorphic embeddings between finite trees.<br> 5.
Borel sets in the plane and graphs of one dimensional Borel
functions.<br> 6. Boolean relations between sets of
integers and their images under integer functions.

It is well known that there are separable Banach spaces without
unconditional basic sequences, and that every Banach space
whose density is bigger than an $\omega$-Erdös cardinal
contains an unconditional basic sequence. We prove that it
is consistent that every Banach space of density bigger than
$\aleph_\omega$ has an unconditional sequence. Consequently,
it is consistent that every Banach space of such density has
a separable quotient.

The proof relies on a Ramsey-like combinatorial property
that $\aleph_\omega$ may have.

This is a joint work with P. Dodos (Athens) and S. Todorcevic
(Toronto).

It is well known that there are separable Banach spaces without
unconditional basic sequences, and that every Banach space
whose density is bigger than an $\omega$-Erdös cardinal
contains an unconditional basic sequence. We prove that it
is consistent that every Banach space of density bigger than
$\aleph_\omega$ has an unconditional sequence. Consequently,
it is consistent that every Banach space of such density has
a separable quotient.

The proof relies on a Ramsey-like combinatorial property
that $\aleph_\omega$ may have.

This is a joint work with P. Dodos (Athens) and S. Todorcevic
(Toronto).

Ilijas Farah
(York)Order property of II_1 factors and its applications

Feb. 18
1:30 p.m.
Room 210

Jocelyn Bell
(Buffalo)The Uniform Box Product Problem

An important unsolved problem in topology is the box product
problem, which asks whether the product of compact spaces
with the box topology is normal. Applying uniformities, we
introduce a new topology on products which sits between the
box and Tychonov products called the uniform box product.
This new product is an extension of the sup metric to products
of compact spaces. We will show, in ZFC, that the uniform
box product of a certain non-metrizable compact space is normal.

A set theorist will find interesting opportunities by turning
to the Vapnik-Chervonenkis theory of statistical learning.
To argue this point, we will discuss the basic concepts and
results of the theory of probably approximately correct (PAC)
learnability, and then proceed to some recent results by the
speaker. Those include a solution of a problem by Vidyasagar
about PAC learnability over non-atomic measures which use
Martin's Axiom and a combinatorial parameter defined for Boolean
algebras. We will also discuss some other results and open
problems.

Nov. 12
1:30 p.m.
Room 210

Juris Steprans
(York)Reflecting non-meagreness in the Erdos-Kakutani group (joint
work with Marton Elekes)

I will discuss the notion of Roelcke precompactness and how
it can be applied to the study of unitary representations.
I plan to give a fairly complete proof of the classification
theorem for the representations of automorphism groups of
omega-categorical structures.

Oct. 22
1:30 p.m.
Room 210

Julien Melleray
(Lyon)Polish topometric groups

Oct. 29
1:30 p.m.
Room 210

Lionel Nguyen
Van Thé (Université Aix-Marseille 3, Paul
Cézanne)Partition properties of the dense local order
(joint with Claude Laflamme and Norbert Sauer, University of
Calgary)

In 1984, Lachlan classified the countable ultrahomogeneous
tournaments (ie the countable directed graphs where every
pair or points supports an arc, and where every isomorphism
between finite subgraphs extends to an automorphism of the
whole structure), and showed that there are only three such
objects: the rationals, the countable random tournament, and
the so-called dense local order. The purpose of this talk
is to present the Ramsey properties of this latter object.

Peter Burton, Kevin Duanmu, Frank Tall (U of T)Lindelof productsA problem in Przymusinski's Handbook survey on normality
of products asks whether, if X x Y is Lindelof for every Lindelof
Y, are all countable powers of X Lindelof. We show that the
Continuum Hypothesis implies a positive answer.

Sep 17
1:30 p.m.
Fields Rm 210

Peter Burton, Kevin Duanmu, Frank Tall (U of T)Lindelof productsA problem in Przymusinski's Handbook survey on normality
of products asks whether, if X x Y is Lindelof for every Lindelof
Y, are all countable powers of X Lindelof. We show that the
Continuum Hypothesis implies a positive answer.

Sep 10
1:30 p.m.
Fields
Rm 210

Franklin Tall, University of TorontoMore applications of PFA(S)[S]

Friday,
Sept. 3

No Seminar

Friday,
August 27

Asger Törnquist
( KCRG Wien)Conjugacy, orbit equivalence and von Neumann equivalence
are analytic.For a countably infinite discrete group G, there are three
equivalence relations of major interest for its measure preserving
ergodic actions: Conjugacy, orbit equivalence and von Neumann
equivalence. These equivalence relations are prima facie analytic.
Can they be Borel?
For G amenable, only conjugacy is of interest. In the special
case G=Z the question goes back to Halmos, but was only recently
solved by Forman, Rudolph and Weiss, who showed that conjugacy
is complete analytic (as a
set.)
In this talk I will show that for a class of non-amenable groups,
including all free groups, that all three of the above equivalence
relations are analytic and not Borel.
Part of this work is joint with Inessa Epstein.

Friday,
August 20

Ilijas Farah (York)A very short introduction to Woodin's Pmax forcing

Friday,
August 13

Paul Larson (Miami University
of Ohio)Fragments of Martin's Maximum in the Pmax extension

Friday,
August 6

Franklin TallPFA(S)[S] and applications cont.We develop machinery for applying the method of forcing
with a coherent Souslin tree S over models of PFA restricted
to posets that preserve S.

Friday,
July 30

Franklin TallPFA(S)[S] and applicationsWe develop machinery for applying the method of forcing
with a coherent Souslin tree S over models of PFA restricted
to posets that preserve S.